3.75.7 \(\int \frac {x^{\frac {4}{-2 x+\log ^4(3+e^{3-x})}} (-24 x-8 e^{3-x} x+(12+4 e^{3-x}) \log ^4(3+e^{3-x})+(24 x+8 e^{3-x} x) \log (x)+16 e^{3-x} x \log ^3(3+e^{3-x}) \log (x))}{12 x^3+4 e^{3-x} x^3+(-12 x^2-4 e^{3-x} x^2) \log ^4(3+e^{3-x})+(3 x+e^{3-x} x) \log ^8(3+e^{3-x})} \, dx\)

Optimal. Leaf size=26 \[ x^{\frac {2}{-x+\frac {1}{2} \log ^4\left (3+e^{3-x}\right )}} \]

________________________________________________________________________________________

Rubi [F]  time = 3.65, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {x^{\frac {4}{-2 x+\log ^4\left (3+e^{3-x}\right )}} \left (-24 x-8 e^{3-x} x+\left (12+4 e^{3-x}\right ) \log ^4\left (3+e^{3-x}\right )+\left (24 x+8 e^{3-x} x\right ) \log (x)+16 e^{3-x} x \log ^3\left (3+e^{3-x}\right ) \log (x)\right )}{12 x^3+4 e^{3-x} x^3+\left (-12 x^2-4 e^{3-x} x^2\right ) \log ^4\left (3+e^{3-x}\right )+\left (3 x+e^{3-x} x\right ) \log ^8\left (3+e^{3-x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x^(4/(-2*x + Log[3 + E^(3 - x)]^4))*(-24*x - 8*E^(3 - x)*x + (12 + 4*E^(3 - x))*Log[3 + E^(3 - x)]^4 + (2
4*x + 8*E^(3 - x)*x)*Log[x] + 16*E^(3 - x)*x*Log[3 + E^(3 - x)]^3*Log[x]))/(12*x^3 + 4*E^(3 - x)*x^3 + (-12*x^
2 - 4*E^(3 - x)*x^2)*Log[3 + E^(3 - x)]^4 + (3*x + E^(3 - x)*x)*Log[3 + E^(3 - x)]^8),x]

[Out]

4*Defer[Int][x^(-1 + 4/(-2*x + Log[3 + E^(3 - x)]^4))/(-2*x + Log[3 + E^(3 - x)]^4), x] + 8*Defer[Int][(x^(4/(
-2*x + Log[3 + E^(3 - x)]^4))*Log[x])/(2*x - Log[3 + E^(3 - x)]^4)^2, x] + 16*E^3*Defer[Int][(x^(4/(-2*x + Log
[3 + E^(3 - x)]^4))*Log[3 + E^(3 - x)]^3*Log[x])/((E^3 + 3*E^x)*(2*x - Log[3 + E^(3 - x)]^4)^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 x^{-1+\frac {4}{-2 x+\log ^4\left (3+e^{3-x}\right )}} \left (\left (e^3+3 e^x\right ) \log ^4\left (3+e^{3-x}\right )+2 \left (e^3+3 e^x\right ) x (-1+\log (x))+4 e^3 x \log ^3\left (3+e^{3-x}\right ) \log (x)\right )}{\left (e^3+3 e^x\right ) \left (2 x-\log ^4\left (3+e^{3-x}\right )\right )^2} \, dx\\ &=4 \int \frac {x^{-1+\frac {4}{-2 x+\log ^4\left (3+e^{3-x}\right )}} \left (\left (e^3+3 e^x\right ) \log ^4\left (3+e^{3-x}\right )+2 \left (e^3+3 e^x\right ) x (-1+\log (x))+4 e^3 x \log ^3\left (3+e^{3-x}\right ) \log (x)\right )}{\left (e^3+3 e^x\right ) \left (2 x-\log ^4\left (3+e^{3-x}\right )\right )^2} \, dx\\ &=4 \int \left (\frac {4 e^3 x^{\frac {4}{-2 x+\log ^4\left (3+e^{3-x}\right )}} \log ^3\left (3+e^{3-x}\right ) \log (x)}{\left (e^3+3 e^x\right ) \left (2 x-\log ^4\left (3+e^{3-x}\right )\right )^2}+\frac {x^{-1+\frac {4}{-2 x+\log ^4\left (3+e^{3-x}\right )}} \left (-2 x+\log ^4\left (3+e^{3-x}\right )+2 x \log (x)\right )}{\left (2 x-\log ^4\left (3+e^{3-x}\right )\right )^2}\right ) \, dx\\ &=4 \int \frac {x^{-1+\frac {4}{-2 x+\log ^4\left (3+e^{3-x}\right )}} \left (-2 x+\log ^4\left (3+e^{3-x}\right )+2 x \log (x)\right )}{\left (2 x-\log ^4\left (3+e^{3-x}\right )\right )^2} \, dx+\left (16 e^3\right ) \int \frac {x^{\frac {4}{-2 x+\log ^4\left (3+e^{3-x}\right )}} \log ^3\left (3+e^{3-x}\right ) \log (x)}{\left (e^3+3 e^x\right ) \left (2 x-\log ^4\left (3+e^{3-x}\right )\right )^2} \, dx\\ &=4 \int \left (\frac {x^{-1+\frac {4}{-2 x+\log ^4\left (3+e^{3-x}\right )}}}{-2 x+\log ^4\left (3+e^{3-x}\right )}+\frac {2 x^{\frac {4}{-2 x+\log ^4\left (3+e^{3-x}\right )}} \log (x)}{\left (2 x-\log ^4\left (3+e^{3-x}\right )\right )^2}\right ) \, dx+\left (16 e^3\right ) \int \frac {x^{\frac {4}{-2 x+\log ^4\left (3+e^{3-x}\right )}} \log ^3\left (3+e^{3-x}\right ) \log (x)}{\left (e^3+3 e^x\right ) \left (2 x-\log ^4\left (3+e^{3-x}\right )\right )^2} \, dx\\ &=4 \int \frac {x^{-1+\frac {4}{-2 x+\log ^4\left (3+e^{3-x}\right )}}}{-2 x+\log ^4\left (3+e^{3-x}\right )} \, dx+8 \int \frac {x^{\frac {4}{-2 x+\log ^4\left (3+e^{3-x}\right )}} \log (x)}{\left (2 x-\log ^4\left (3+e^{3-x}\right )\right )^2} \, dx+\left (16 e^3\right ) \int \frac {x^{\frac {4}{-2 x+\log ^4\left (3+e^{3-x}\right )}} \log ^3\left (3+e^{3-x}\right ) \log (x)}{\left (e^3+3 e^x\right ) \left (2 x-\log ^4\left (3+e^{3-x}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.19, size = 24, normalized size = 0.92 \begin {gather*} x^{-\frac {4}{2 x-\log ^4\left (3+e^{3-x}\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^(4/(-2*x + Log[3 + E^(3 - x)]^4))*(-24*x - 8*E^(3 - x)*x + (12 + 4*E^(3 - x))*Log[3 + E^(3 - x)]^
4 + (24*x + 8*E^(3 - x)*x)*Log[x] + 16*E^(3 - x)*x*Log[3 + E^(3 - x)]^3*Log[x]))/(12*x^3 + 4*E^(3 - x)*x^3 + (
-12*x^2 - 4*E^(3 - x)*x^2)*Log[3 + E^(3 - x)]^4 + (3*x + E^(3 - x)*x)*Log[3 + E^(3 - x)]^8),x]

[Out]

x^(-4/(2*x - Log[3 + E^(3 - x)]^4))

________________________________________________________________________________________

fricas [A]  time = 0.62, size = 21, normalized size = 0.81 \begin {gather*} x^{\frac {4}{\log \left (e^{\left (-x + 3\right )} + 3\right )^{4} - 2 \, x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*exp(3-x)+12)*log(exp(3-x)+3)^4+16*x*exp(3-x)*log(x)*log(exp(3-x)+3)^3+(8*x*exp(3-x)+24*x)*log(x)
-8*x*exp(3-x)-24*x)*exp(4*log(x)/(log(exp(3-x)+3)^4-2*x))/((x*exp(3-x)+3*x)*log(exp(3-x)+3)^8+(-4*x^2*exp(3-x)
-12*x^2)*log(exp(3-x)+3)^4+4*x^3*exp(3-x)+12*x^3),x, algorithm="fricas")

[Out]

x^(4/(log(e^(-x + 3) + 3)^4 - 2*x))

________________________________________________________________________________________

giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*exp(3-x)+12)*log(exp(3-x)+3)^4+16*x*exp(3-x)*log(x)*log(exp(3-x)+3)^3+(8*x*exp(3-x)+24*x)*log(x)
-8*x*exp(3-x)-24*x)*exp(4*log(x)/(log(exp(3-x)+3)^4-2*x))/((x*exp(3-x)+3*x)*log(exp(3-x)+3)^8+(-4*x^2*exp(3-x)
-12*x^2)*log(exp(3-x)+3)^4+4*x^3*exp(3-x)+12*x^3),x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

maple [A]  time = 0.04, size = 22, normalized size = 0.85

method	result	size
risch	\(x^{\frac {4}{\ln \left ({\mathrm e}^{3-x}+3\right )^{4}-2 x}}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*exp(3-x)+12)*ln(exp(3-x)+3)^4+16*x*exp(3-x)*ln(x)*ln(exp(3-x)+3)^3+(8*x*exp(3-x)+24*x)*ln(x)-8*x*exp(3
-x)-24*x)*exp(4*ln(x)/(ln(exp(3-x)+3)^4-2*x))/((x*exp(3-x)+3*x)*ln(exp(3-x)+3)^8+(-4*x^2*exp(3-x)-12*x^2)*ln(e
xp(3-x)+3)^4+4*x^3*exp(3-x)+12*x^3),x,method=_RETURNVERBOSE)

[Out]

x^(4/(ln(exp(3-x)+3)^4-2*x))

________________________________________________________________________________________

maxima [B]  time = 0.56, size = 65, normalized size = 2.50 \begin {gather*} x^{\frac {4}{x^{4} - 4 \, x^{3} \log \left (e^{3} + 3 \, e^{x}\right ) + 6 \, x^{2} \log \left (e^{3} + 3 \, e^{x}\right )^{2} + \log \left (e^{3} + 3 \, e^{x}\right )^{4} - 2 \, {\left (2 \, \log \left (e^{3} + 3 \, e^{x}\right )^{3} + 1\right )} x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*exp(3-x)+12)*log(exp(3-x)+3)^4+16*x*exp(3-x)*log(x)*log(exp(3-x)+3)^3+(8*x*exp(3-x)+24*x)*log(x)
-8*x*exp(3-x)-24*x)*exp(4*log(x)/(log(exp(3-x)+3)^4-2*x))/((x*exp(3-x)+3*x)*log(exp(3-x)+3)^8+(-4*x^2*exp(3-x)
-12*x^2)*log(exp(3-x)+3)^4+4*x^3*exp(3-x)+12*x^3),x, algorithm="maxima")

[Out]

x^(4/(x^4 - 4*x^3*log(e^3 + 3*e^x) + 6*x^2*log(e^3 + 3*e^x)^2 + log(e^3 + 3*e^x)^4 - 2*(2*log(e^3 + 3*e^x)^3 +
1)*x))

________________________________________________________________________________________

mupad [B]  time = 6.82, size = 26, normalized size = 1.00 \begin {gather*} \frac {1}{x^{\frac {4}{2\,x-{\ln \left ({\mathrm {e}}^{-x}\,{\mathrm {e}}^3+3\right )}^4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-(4*log(x))/(2*x - log(exp(3 - x) + 3)^4))*(log(exp(3 - x) + 3)^4*(4*exp(3 - x) + 12) - 8*x*exp(3 - x
) - 24*x + log(x)*(24*x + 8*x*exp(3 - x)) + 16*x*exp(3 - x)*log(exp(3 - x) + 3)^3*log(x)))/(log(exp(3 - x) + 3
)^8*(3*x + x*exp(3 - x)) - log(exp(3 - x) + 3)^4*(4*x^2*exp(3 - x) + 12*x^2) + 4*x^3*exp(3 - x) + 12*x^3),x)

[Out]

1/x^(4/(2*x - log(exp(-x)*exp(3) + 3)^4))

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*exp(3-x)+12)*ln(exp(3-x)+3)**4+16*x*exp(3-x)*ln(x)*ln(exp(3-x)+3)**3+(8*x*exp(3-x)+24*x)*ln(x)-8
*x*exp(3-x)-24*x)*exp(4*ln(x)/(ln(exp(3-x)+3)**4-2*x))/((x*exp(3-x)+3*x)*ln(exp(3-x)+3)**8+(-4*x**2*exp(3-x)-1
2*x**2)*ln(exp(3-x)+3)**4+4*x**3*exp(3-x)+12*x**3),x)

[Out]

Timed out

________________________________________________________________________________________